On $p$-modulus estimates in Orlicz-Sobolev classes

TL;DR

This paper investigates the properties of homeomorphisms with finite distortion in Orlicz-Sobolev classes, establishing conditions under which they are lower Q-homeomorphisms and ring Q-homeomorphisms related to p-modulus, with implications for geometric function theory.

Contribution

It provides new conditions under which homeomorphisms in Orlicz-Sobolev spaces are classified as lower Q- and ring Q-homeomorphisms based on p-modulus estimates.

Findings

Homeomorphisms with finite distortion in W^{1,φ}_{loc} are lower Q-homeomorphisms under Calderon-type conditions.

Such homeomorphisms are also ring Q-homeomorphisms with respect to specific p-modulus.

Results connect Orlicz-Sobolev regularity with geometric distortion properties.

Abstract

Under a condition of the Calderon type on $\varphi$, we show that a homeomorphism $f$ of finite distortion in $W^{1,\varphi}_{\rm loc}$ and, in particular, $f\in W^{1,q}_{\rm loc}$ for $q>n-1$ in ${\Bbb R}^n$, $n\ge 3$, is a lower $Q$-homeomorphisms with respect to the $p$-modulus with $\left[ K_{I,\alpha}(x,f)\right]^{\beta}$, $p>n-1$ and a ring $Q_{*}$-homeomorphism with respect to the $\frac{p}{p-n+1}$-modulus with $Q_{*}(x)=K_{I,\alpha}(x,f)$ where $ K_{I,\alpha}(x,f)$ is its inner $\alpha$-dilatation and $\alpha=\frac{p}{p-n+1}$, $\beta=\frac{p-n+1}{n-1}$.